Abstract
There are two main goals in this research, the first one is presenting an extended Poincaré–Lighthill–Kuo method (EPLKM) for studying the collisions between two dark envelope solitons in a non-Maxwellian plasma permeated by a positron beam, drawing inspiration from recent laboratory experiments on positron beams. The second goal in the present model explains the method of lines (MOLs) approach devoted for analyzing and modeling freak waves (FWs) . In order to achieve the goals of this study, a nonlinear Schrödinger equation (NLSE) has been derived using a reductive perturbation technique (RPT). The regions of (un)stable structures (dark and bright solitons and FWs) have been precisely defined based on the studying of modulational instability (MI) of the NLSE. In the stable regions, we focus our attention in examining and investigating dark soliton collisions. To do that a counterpart pair of two dark envelope solitons is assumed as initial condition, and EPLKM is employed to get expressions for the analytical phase shifts and determine the soliton trajectories after collisions. A parametric investigation is also presented by the effect of the superthermality (kappa) parameter and the positron beam characteristics (concentration, streaming velocity) on the colliding soliton properties. On the other side the MOLs approach is introduced for studying the rogue wave (RW) solution of the NLSE in the unstable regions. The numerical results of the comparison between the numerical and analytical solutions showed the accuracy of the MOLs. What prompted us to use this method is its high accuracy and ease of use rather than more complex numerical methods. These results will be helpful in understanding the dynamics of modulated structures (envelope dark soliton collisions and FWs) in purpose designed experiments.
•The PLK method is devoted for studying the dark soliton collisions in a non-Maxwellian plasmas.•The fluid govern equations of a positron beam plasma are reduced to the NLSE.•In the stable regions of modulated structures, the dark soliton collisions have been investigated.•In the unstable regions of modulated structures, the RWs have been discussed.•The MOLs approach is introduced for modelling RWs.•The comparison between the numerical and analytical RW solutions has been carried out.